A process for the cryogenic distillation of an air feed to produce an ultra-high purity oxygen product wherein an oxygen-containing but heavy contaminants-lean (free) stream is removed from the main distillation column system and subsequently stripped in an auxiliary distillation column is taught in the art. Specifically, U.S. Pat. No. 5,049,173 by Cormier et al. teaches such a process. A key feature in Cormier is the method of providing heat duty/reboil to the bottom of the auxiliary distillation column. Where the main distillation column system in Cormier comprises a single distillation column, Cormier's method of providing heat duty/reboil to the bottom of the auxiliary distillation column consists of partially condensing a portion of the gaseous nitrogen overhead. On the other hand, where the main distillation column system in Cormier comprises the classical high pressure/low pressure column arrangement, Cormier's heat duty method consists of at least partially condensing a portion of the "intermediate" gaseous nitrogen overhead from the high pressure column and/or subcooling a portion of the "intermediate" liquid oxygen bottoms from the high pressure column. There is a concern with Cormier, however, in that such methods may not be the most efficient way of providing heat duty/reboil to the bottom of the auxiliary distillation column. It is an object of the present invention to more efficiently provide heat duty/reboil to the bottom of the auxiliary distillation column and thereby more efficiently produce the ultra-high purity oxygen product.